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The plane is enough because all the vectors – before and after the operations (which are really simple rotations) – belong to a two-dimensional plane. The Hilbert space has many more dimensions but they're orthogonal to the plane of the picture and the coordinates (amplitudes) in these additional directions don't change during the calculation so we don't need to draw them at all. The rotation only mixes two coordinates – if the basis is correctly chosen (the plane is a different plane than some plane generated by two "a priori" basis vectors but it exists, anyway).